The exercise of an American option and the surrender of a life insurance contract result both in no payments beyond the date of exercise or surrender, respectively.
In these cases the theory of optimal stopping would be adequate, and we would not have to introduce general theory of impulse control. Now we replace the surrender option of a life insurance contract introduced in Section 4.5 by a free policy option.
A free policy option is an option to subscribe all future premium payments to zero against a corresponding subscription of all future benefits. What could be meant by a corresponding subscription will be explained below. After the time of conversion
4.6. THE FREE POLICY OPTION IN LIFE INSURANCE 97 to free policy, the contract is not stopped but continues with converted payments and therefore general theory of impulse control applies. The free policy option could be considered in combination with the surrender option. Then one could allow for surrender of a policy before or after conversion into free policy or both. However, in order to keep things relatively simple we choose to disregard the surrender option in this section and take into account the free policy option exclusively.
We consider once more the classical multi-state life insurance policy described in Section 4.5, now with the free policy option. To handle this situation we introduce, as in Section 4.5, a third non-marketed indexS2, now indicating whether the contract is converted or not. We let
dSt2 =dNtI, S02 = 0.
whereNtI counts the number of conversions until time t(equals 0 or 1). We assume that no payments fall due on transition of S2 from 0 to 1, but now a free policy reserve has to be set aside. This will be a reserve for subscribed benefits. Since these subscribed payments, to be defined below, will depend on the time elapsed since conversion, we need a fourth index measuring this duration,
dSt3 =dt−St3−dNtI, S03 = 0.
Note that as long asSt2 = 0, we know thatSt3 =t, the time elapsed since the issue of the contract. For all processes below, we abbreviate the argument (t, St0, St1, St2, St3) by a subscript t and an argument (St2, St3), such that we e.g. can write Vt(St2, St3) instead ofV (t, St0, St1, St2, St3). Then the fact that no payments fall due on transition of S2 from 0 to 1 is written
b1t(0, t) = 0.
Let Bb be the payment process given that the insurance contract is not yet con- verted, i.e. dBt(0, t) = dBbt. We emphasize that this notation has no direct con- nection with first order payments introduced in Chapter 3. Now, we assume that the benefits are subscribed proportionally with a proportionality factor given by the ratio between the technical reserve and the technical free policy reserve at the conversion time. Then
dBt 1, St3
=dBbt−Vt∗−S3 t
Vt∗−−S3 t
, (4.25)
where
dBbt− = dBbt
− , Vt∗−S3
t = EQ∗
"Z T t−St3
Zt0−∗S3 t
Zs0∗ d
−Bbs St1−S3
t
# , Vt∗−−S3
t = EQ∗
"Z T t−St3
Zt0−∗S3 t
Zs0∗ d
−Bbs− St1−S3
t
# ,
and, accordingly,
Vt 1, St3
= EQ Z T
t
Zt0
Zs0d −Bs 1, St3 St
= Vt∗−S3 t
Vt∗−−S3 t
Vt−,
where
Vt−=EQ Z T
t
Zt0 Zs0d
−Bbs− St
.
From Theorem 4 we get that for an arbitrage free contract the reserve, and hereby the single premium B0 (or another balancing element of B), is given by a regular solution to the variational inequality (4.22) withVt∗replaced byVt(1,0), the reserve for free policy benefits at the time of conversion.
Now we only need a differential system for calculation of the free policy reserve Vt(1, St3). Since no intervention is allowed for in a free policy, we see from Theorem 3 that for an arbitrage free contract, the free policy reserve Vt(1, St3) is determined by a regular solution to the differential equation (Vt=V (t, s0, s1,1, s3)),
∂tVt = bct+rtVt− bdt +VtJ −Vt1×J
àt−∂s3Vt, (4.26) VT− = ∆BT.
By (4.22) with Vt∗ replaced by Vt(1,0) and Vt(1, St3) given by (4.26), we now have a constructive tool for determination of the reserve Vt.
If the free policy payments are given by (4.25), we can draw conclusions from Theorem 5. Now, Yt in (4.18) is given by
Yt= Z t
ρ
Zρ0
Zs0d(−Bs(0, s)) + Zρ0
Zt0Vt(1,0),
such that Ito’s lemma and (4.6) with (r, à) replaced by (r∗, à∗) gives Zt0
Zρ0dYt = d(−Bt(0, t))−rt(1,0)Vt(1,0) +dVt(1,0)
= d(−Bt(0, t))−rt(1,0)Vt(1,0) + Vt∗J (1,0)−Vt∗1×J(1,0) dNt
+ (bct(1,0) +rt(1,0)Vt(1,0))dt
− bdt(1,0) +VtJ(1,0)−Vt1×J(1,0)
àt(1,0)−∂s3Vt(1,0) dt
= d(Bt(1,0)−Bt(0, t))−∂s3Vt(1,0)dt+Rt(1,0)dMtQ, (4.27) with
Rt(1,0) =bdt(1,0) +VtJ (1,0)−Vt1×J(1,0).
4.6. THE FREE POLICY OPTION IN LIFE INSURANCE 99 For the first term of (4.27), we have
d(Bt(1,0)−Bt(0, t)) = Vt∗
Vt∗−dBbt−−dBbt
=
Vt∗ Vt∗−
bbct−−bbdt−àt
−bbct+bbdt−àt
dt +
−Vt∗−
Vt∗−− bbdt−−+bbdt−
dMtQ, (4.28)
and, introducing
R∗t = bbdt +Vt∗J −Vt∗1×J, R−t = bbdt−+Vt−J −Vt−1×J, R∗−t = bbdt−+Vt∗−J −Vt∗−1×J, we have for the second term of (4.27),
∂s3Vt(1,0) = Vt−
Vt∗−∂s3Vt∗− Vt−Vt∗
Vt∗−2∂s3Vt∗−
= −Vt−
Vt∗−∂tVt∗ + Vt−Vt∗
Vt∗−2∂s3Vt∗−
= −Vt− Vt∗−
bbct+r∗tVt∗−R∗tà∗t dt + Vt−Vt∗
Vt∗−2
bbct−+r∗tVt∗−−R∗−t àt dt
= −Vt− Vt∗−
bbct−R∗tà∗t
+ Vt−Vt∗ Vt∗−2
bbct−−R∗−t àt!
dt. (4.29) Then, by (4.27), (4.28), and (4.29), we get
Zt0
Zρ0dYt =
"
Vt− Vt∗−
bbct −R∗tà∗t
− Vt−Vt∗ Vt∗−2
bbct−−R∗−t à∗t
+ Vt∗ Vt∗−
bbct−−bbdt−àt
−bbct+bbdtàt
dt +
Rt−(1,0)− Vt∗−
Vt∗−− bbdt−−+bbdt−
dMtQ
=
Vt−
Vt∗− −1 bbct− Vt∗ Vt∗−bbct−−
bbdt − Vt∗ Vt∗−bbdt−
àt
(4.30)
−Vt− Vt∗−
Vt∗J − Vt∗ Vt∗−Vt∗−J
àt +Vt−
Vt∗−
R∗t − Vt∗ Vt∗−R∗−t
(àt−à∗t)
dt +
Rt−(1,0)− Vt∗−
Vt∗−− bbdt−−+bbdt−
dMtQ
We are interested in the drift term of ZZ0t0
ρdYt in order to draw conclusions from Theorem 5. It seems difficult to come up with general conditions for r∗ and à∗ but we shall take a closer look at the drift term in a special case.
An example
We shall work with a simple insurance contract as an illustration of the drift term of (4.30). The insurance contract is a single life endowment insurance with a sum insured of 1 and a constant premiumπ as long as the insured is alive. Let X be the two-state process defined by Xt = 0 if the insured is alive at time t, Xt = 1 if the insured is dead at time t, and X0 = 0. Then
αSt =
rtSt0 0
, βSt = 0
1
,s0 = 1
0
. Given that St1 = 0, we have
bbct = π, bbct− = 0,
bbdt = bbdt− = 1, such that the drift term of ZZt00
ρdYt, (4.30), given St1 = 0, equals Vt−
Vt∗− −1 π−
1− Vt∗ Vt∗−
àt
+ Vt− Vt∗−
1− Vt∗ Vt∗−
(àt−à∗t). Introducing traditional actuarial notation, one can write this drift term as
Ax+t T−t| A∗x+t T−t| −1
!
π− 1− A∗x+t T−t|−πa∗x+t T−t| A∗x+t T−t|
! àt
!
+Ax+t T−t|
A∗x+t T−t| 1− A∗x+t T−t|−πa∗x+t T
−t|
A∗x+tT−t|
!
(àt−à∗t)
= Ax+t T−t| A∗x+t T−t| −1
! π A∗x+t T−t|
T−tEx+t∗ + Z T
t
a∗s−t|s−tp∗x+t à∗x+s−àx+t ds
+Ax+t T−t|πa∗x+t T
−t|
A∗x+t T−t|2 (àt−à∗t).
We are interested in sufficient conditions for this quantity to be non-positive such that Y is a Q-supermartingale. Term by term, it can be seen that this will be the case if à∗ > à, r > r∗, and à∗ is increasing. According to Theorem 5, it is now optimal to convert the into a free policy immediately and the insurance company should set aside the free policy reserve, Vt(1,0).
It is worth noting that these conditions onà∗ andr∗ also would impose optimality of immediate surrender in the case of the surrender option: If à∗ is increasing, Vt∗ and Rt∗ are non-negative for allt ≤T. Now, this conclusion follows from (4.24).
Part III
Control in life and pension insurance
101
Chapter 5
Risk-adjusted utility
This chapter introduces an idea of risk-adjusted utility. Instead of measuring moral value or utility of nominal value, we suggest to measure utility of deflated value.
As deflator is chosen the same deflator which is used for determination of price or financial value. Using this concept we study the problem of optimal investment- consumption and the problem of utility indifference pricing in an incomplete market.